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Theorem dcned 2261
Description: Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
Hypothesis
Ref Expression
dcned.eq (𝜑DECID 𝐴 = 𝐵)
Assertion
Ref Expression
dcned (𝜑DECID 𝐴𝐵)

Proof of Theorem dcned
StepHypRef Expression
1 dcned.eq . . 3 (𝜑DECID 𝐴 = 𝐵)
2 dcn 784 . . 3 (DECID 𝐴 = 𝐵DECID ¬ 𝐴 = 𝐵)
31, 2syl 14 . 2 (𝜑DECID ¬ 𝐴 = 𝐵)
4 df-ne 2256 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
54dcbii 785 . 2 (DECID 𝐴𝐵DECID ¬ 𝐴 = 𝐵)
63, 5sylibr 132 1 (𝜑DECID 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 780   = wceq 1289  wne 2255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781  df-ne 2256
This theorem is referenced by:  nn0n0n1ge2b  8796  algcvgblem  11124
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